This number tells the energy level of electrons in that shell (although there's other factors).
[citation needed] Accounting for two states of spin, each n-shell can accommodate up to 2n2 electrons.
In a simplistic one-electron model described below, the total energy of an electron is a negative inverse quadratic function of the principal quantum number n, leading to degenerate energy levels for each n > 1.
Description of energy levels based on n alone gradually becomes inadequate for atomic numbers starting from 5 (boron) and fails completely on potassium (Z = 19) and afterwards.
The principal quantum number was first created for use in the semiclassical Bohr model of the atom, distinguishing between different energy levels.
With the development of modern quantum mechanics, the simple Bohr model was replaced with a more complex theory of atomic orbitals.
Two electrons belonging to the same atom cannot have the same values for all four quantum numbers, due to the Pauli exclusion principle.
The principal quantum number arose in the solution of the radial part of the wave equation as shown below.
The concept of energy levels and notation were taken from the earlier Bohr model of the atom.
Schrödinger's equation developed the idea from a flat two-dimensional Bohr atom to the three-dimensional wavefunction model.
This formula is not correct in quantum mechanics as the angular momentum magnitude is described by the azimuthal quantum number, but the energy levels are accurate and classically they correspond to the sum of potential and kinetic energy of the electron.
The principal quantum number n represents the relative overall energy of each orbital.
The minimum energy exchanged during any wave–matter interaction is the product of the wave frequency multiplied by the Planck constant.
This causes the wave to display particle-like packets of energy called quanta.
In the notation of the periodic table, the main shells of electrons are labeled: based on the principal quantum number.
The definite total energy for a particle motion in a common Coulomb field and with a discrete spectrum, is given by:
This discrete energy spectrum resulted from the solution of the quantum mechanical problem on the electron motion in the Coulomb field, coincides with the spectrum that was obtained with the help application of the Bohr–Sommerfeld quantization rules to the classical equations.
[2] In chemistry, values n = 1, 2, 3, 4, 5, 6, 7 are used in relation to the electron shell theory, with expected inclusion of n = 8 (and possibly 9) for yet-undiscovered period 8 elements.
In atomic physics, higher n sometimes occur for description of excited states.
Observations of the interstellar medium reveal atomic hydrogen spectral lines involving n on order of hundreds; values up to 766[3] were detected.